Internal problem ID [6443]
Internal file name [OUTPUT/5691_Sunday_June_05_2022_03_47_13_PM_63020299/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 4. Power Series Solutions and Special Functions. Section 4.4. REGULAR
SINGULAR POINTS. Page 175
Problem number: 1(c).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_ode_missing_y", "second order series method. Irregular singular point"
Maple gives the following as the ode type
[[_2nd_order, _missing_y]]
Unable to solve or complete the solution.
\[ \boxed {x^{2} y^{\prime \prime }+\left (2-x \right ) y^{\prime }=0} \] With the expansion point for the power series method at \(x = 0\).
The type of the expansion point is first determined. This is done on the homogeneous part of the ODE. \[ x^{2} y^{\prime \prime }+\left (2-x \right ) y^{\prime } = 0 \] The following is summary of singularities for the above ode. Writing the ode as \begin {align*} y^{\prime \prime }+p(x) y^{\prime } + q(x) y &=0 \end {align*}
Where \begin {align*} p(x) &= -\frac {x -2}{x^{2}}\\ q(x) &= 0\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([\infty ]\)
Irregular singular points : \([0]\)
Since \(x = 0\) is not an ordinary point, then we will now check if it is a regular singular point. Unable to solve since \(x = 0\) is not regular singular point. Terminating.
Verification of solutions N/A
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y <- LODE missing y successful`
✗ Solution by Maple
Order:=8; dsolve(x^2*diff(y(x),x$2)+(2-x)*diff(y(x),x)=0,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.027 (sec). Leaf size: 64
AsymptoticDSolveValue[x^2*y''[x]+(2-x)*y'[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 e^{2/x} \left (\frac {2835 x^7}{2}+315 x^6+\frac {315 x^5}{4}+\frac {45 x^4}{2}+\frac {15 x^3}{2}+3 x^2+\frac {3 x}{2}+1\right ) x^3+c_1 \]